The benefits of money transfers by MoneyGram: there is no need to open a checking account when implementing the transfer. In due time MoneyGram separated from the Western Union and became an independent brand. Since going public two and a half years ago, MoneyGram has made great strides by pouring energy and resources into several critically important areas for business. How quickly the money will reach the recipient? MoneyGram Money Order comes anywhere in the world in 10 minutes. Until 2004, the owners of MoneyGram were Viad Corp and Travelers Express Company Inc.